# Definitions/ intuition for differential forms

I've read about differential forms in the Princeton Companion to Mathematics by Gower and in baby Rudin and I'm having trouble reconciling the two expositions.

Rudin says a k-form in $E$ ($\subset R^n$) is a function $\omega$, symbolically represented by the sum $$\omega = \sum{a_{i_1 \cdots i_k}(x) dx_{i_1}\wedge \cdots \wedge dx_{i_k}}$$

(where the indices $i_1, \ldots, i_k$ range independently from 1 to n), which assigns to each k-surface $\phi$ in $E$ a number $\omega(\phi) = \int_{\phi}{\omega}$ according to the rule $$\int_{\phi}{\omega}= \int_{D}{\sum{a_{i_1 \dots i_k}(\phi(u)) \frac{\partial (x_{i_1}, \ldots, x_{i_k})}{\partial (u_1, \ldots, u_k)} du}},$$

where $D$ is the (compact) domain of $\phi$ (for instance the k-cell $[0,1]^k$) and $\frac{\partial (x_{i_1}, \ldots, x_{i_k})}{\partial (u_1, \ldots, u_k)}$ is the Jacobian.

Meanwhile the article by Tao in the Princeton Companion on differential forms gave me the following understanding: a k-form continuously assigns to each point in $R^n$ a (linear) functional which takes in an infinitesimal k-dimensional parallelepiped with dimensions $\Delta x_1 \wedge \cdots \wedge \Delta x_k$ and returns a scalar. Thus if we evaluate a k-form $\omega$ at some point $x$ we get the linear functional $$\omega_x : \wedge^k (R^n) \rightarrow R.$$ So if $\omega$ is a 0-form then $\omega_x$ sends scalars to scalars, if its a 1-form then it sends (infinitesimal) vectors to scalars, if its a 2-form it sends (infinitesimal) parallelograms to scalars, etc. Finally to integrate a k-form against a k-surface we split up the k-surface into infinitesimal k-parallelepipeds, evaluate the k-form at each base point, plug in the associated k-parallelepiped into the resulting functional, and sum across the entire k-surface. And this process is precisely what Rudin's integral $\int_{\phi}{\omega}$ does.

Please let me know if my understanding is correct. I would greatly appreciate any suggestions about how to better understand differential forms.

• Rudin defines forms in a rather odd way to avoid huge prerequisites (differential manifolds, tangent bundles, tensors). It's a tour de force but a lot of math students will eventually acquire the prerequisites and see forms in a somewhat different light, closer to your description of Tao's treatment (which I'm unfamiliar with). Caveat: the "infinitesimals" you speak of are unknown to me. For a more standard treatment of differential forms there is Lee's Smooth Manifolds, for example. – ForgotALot Dec 23 '15 at 6:51